Falling plumes in bacterial bioconvection

Experiments by Kessler on bioconvection in laboratory suspensions of bacteria (Bacillus subtilis), contained in a deep chamber, reveal the development of a thin upper boundary layer of cell-rich fluid which becomes unstable, leading to the formation of falling plumes. We use the continuum description of such a suspension developed by Hillesdon et at. (1995) as the basis for a theoretical model of the boundary layer and an axisymmetric plume. If the boundary layer has dimensionless thickness lambda much less than 1, the plume has width lambda (1/2). A similarity solution is found for the plume in which the cell flux and volume flux can be matched to those in the boundary layer and in the bulk of the suspension outside both regions. The corresponding model for a two-dimensional plume fails to give a self-consistent solution.